Non-intrusive Polynomial Chaos Method Research Articles

Copula-based Bayesian uncertainty quantification framework of SST turbulence model for flow over a Gaussian bump

Quantitatively analyze the uncertainties in turbulence models is of great meaning for research and engineering. In this paper, we develop a Bayesian uncertainty analysis framework based on copulas. The proposed framework is applied to the parametric uncertainty quantification and calibration of the shear-stress transport turbulence model for the flow over a Gaussian bump. A copula function is used to establish the correlation between variables and construct the weight function, which enables greater consideration of Bayesian inference. Prior analysis is conducted based on Sobol indices with the nonintrusive polynomial chaos method, demonstrating the important effects of κ, a1, and β∗ on the flow, both for the wall force coefficients and the Reynolds shear stress. Next, the wall force coefficients and Reynolds shear stress are used as training sets to verify the effectiveness of the method. The results show that modeling the relationship based on copulas is necessary and effective, and that the weight function adaptively balances the influence of physical quantities at different inference locations. The proposed method effectively corrects the computational deviations of the corresponding physical quantities. In addition, the confidence interval for the posterior sample is increased, enhancing the likelihood of obtaining results close to the actual values.

Analysis of geometric uncertainties in 3D thermo-fluid problems solved by RBF-FD meshless method

The tolerances of production processes can lead to uncertainties in the behaviour and in the features of the manufactured products. From the point of view of the design of engineering components it is therefore of valuable practical interest to be able to quantify such uncertainties as well as the expected, i.e., averaged, performances. Such uncertainty quantification is carried out in this work by means of the Non-Intrusive Polynomial Chaos (PC) method in order to estimate the propagation of geometrical uncertainties of the boundaries, i.e., when the boundaries are described by stochastic variables. Existing deterministic solvers can be used with the PC method because of its non-intrusive formulation, allowing an accurate and practical prediction of the random response through a simple set of deterministic response simulations. The Radial Basis Function Finite Differences (RBF-FD) method is employed as a black box solver for the computation of the required set of responses defined over deterministic boundaries. The RBF-FD method belongs to the class of meshless methods which do not require a computational mesh/grid, therefore its main capability is to easily deal with practical problems defined over complex-shaped domains. The geometrical flexibility of the RBF-FD method is even more advantageous when coupled to the Non-Intrusive PC method for uncertainty quantification since different deterministic solutions over different geometries are required. The applicability of the proposed approach to practical problems is presented through the prediction of geometric uncertainty effects for a steady-state forced convection problem in a 3D complex-shaped domain.

Accelerating model evaluations in uncertainty propagation on tensor grids using computational graph transformations

Methods such as non-intrusive polynomial chaos (NIPC), and stochastic collocation are frequently used for uncertainty propagation problems. Particularly for low-dimensional problems, these methods often use a tensor-product grid for sampling the space of uncertain inputs. A limitation of this approach is that it encounters a significant challenge: the number of sample points grows exponentially with the increase of uncertain inputs. Current strategies to mitigate computational costs abandon the tensor structure of sampling points, with the aim of reducing their overall count. Contrastingly, our investigation reveals that preserving the tensor structure of sample points can offer distinct advantages in specific scenarios. Notably, by manipulating the computational graph of the targeted model, it is feasible to avoid redundant evaluations at the operation level to significantly reduce the model evaluation cost on tensor-grid inputs. This paper presents a pioneering method: Accelerated Model Evaluations on Tensor grids using Computational graph transformations (AMTC). The core premise of AMTC lies in the strategic modification of the computational graph of the target model to algorithmically remove the repeated evaluations on the operation level. We implemented the AMTC method within the compiler of a new modeling language called the Computational System Design Language (CSDL). We demonstrate the effectiveness of AMTC by using it with the full-grid NIPC method to solve four low-dimensional UQ problems involving an analytical piston model, a multidisciplinary unmanned aerial vehicle design model, a multi-point air taxi mission analysis model, and a single-disciplinary rotor model, respectively. For three of the four test problems, AMTC reduces the model evaluation cost by between 50% and 90%, making the full-grid NIPC the most efficacious method to use among the UQ methods implemented.

Intrusive and non-intrusive uncertainty quantification methodologies for pyrolysis modeling

In this work we discuss and compare multiple uncertainty analysis methodologies for pyrolysis modeling. Using Arrhenius equation as the pyrolysis model and kinetic parameters for PMMA material as uncertain variables, different approaches for stochastic uncertainty quantification and their computational effort are analyzed. First non-intrusive (NI) methods are compared through convergence analysis where the deterministic model can be used as is to ascertain the statistical moments, followed by an intrusive polynomial chaos (IPC) approach which requires reformulation of the deterministic model, but solution of the model only once. The applied NI methods include Monte Carlo based simulations, regression and projection based non-intrusive polynomial chaos (NIPC) methods. The uncertainty analysis and convergence comparison is performed for three materials including the effects of overlapping reactions and dependent parameters. In general for different uncertain parameters and materials, IPC gave accurate statistical information in a single model run compared to NIPC projection requiring runs ranging from tens to hundreds and NIPC regression requiring runs in range of thousands to generate converged statistics. However, in case of only a few uncertain parameters, presence of non-polynomial functions in the model and small-scale simulations, projection based NIPC outshines IPC in terms of simplicity and applicability.

Non-linear stochastic dynamics analysis of mechanical systems using non-intrusive polynomial chaos method: application to pole vaulting

Non-linear stochastic dynamics analysis of mechanical systems using non-intrusive polynomial chaos method: application to pole vaulting

Uncertainty and Sensitivity Study on Lateral Jet Interaction for Hypersonic Missile

The demanding working environment of hypersonic missiles imposes high requirements on the reliability and stability of the reaction control system. This work aims to investigate the impact of uncertainties in freestream and jet parameters on the flowfield and aerodynamic performance of a missile with a lateral jet. Initially, the influence of angle of attack and Mach number on aerodynamic characteristics is evaluated. The findings indicate that the lateral jet control effect is enhanced as the Mach number increases. Moreover, the interference force/moment does not vary linearly with the angle of attack, and there are notable differences in the flow characteristics when the jet is on the windward and leeward sides. Subsequently, six parameters are selected as input variables. The uncertainties of the output results are analyzed using a stochastic expansion based on the nonintrusive polynomial chaos method. Sensitivity analysis is performed using Sobol indices to estimate the relative contributions of each parameter to the overall uncertainty. Within the given uncertainty interval, the surface pressure of the missile exhibits significant variation, with a maximum uncertainty of approximately 480%. The uncertainty of the recirculation zone range is 50.14%, while the pitch moment amplification factor has an uncertainty of 52.44%. The freestream Mach number, angle of attack, and jet pressure ratio are identified as key factors affecting the aerodynamic performance, surface pressure coefficient, and jet penetration height. Conversely, the contribution of other parameters can be considered negligible.

Inferring displacement fields from sparse measurements using the statistical finite element method

Nowadays, strain and displacement can be measured using techniques such as electronic speckle pattern interferometry and digital image correlation. However, usually, only some part of the domain of interest is accessible for measurement devices. In order to assess also inaccessible areas of the structure, a fundamental task for structural health monitoring is the inference of full-field displacements from sparse measurements. Another important task is the inference of mechanical stress from displacement or strain data. In computational mechanics, the link between such data and stress is established via constitutive models.A well-established approach for inferring full displacement and stress fields from possibly sparse data is to calibrate the parameter of a given constitutive model using a Bayesian update. After calibration, a (stochastic) forward simulation is conducted with the identified model parameters to resolve physical fields in regions that were not accessible to the measurement device. A shortcoming of model calibration is that the model is deemed to best represent reality, which is only sometimes the case, especially in the context of the aging of structures and materials. While this issue is often addressed with repeated model calibration, a different approach is followed in the recently proposed statistical Finite Element Method (statFEM). Instead of using Bayes’ theorem to update model parameters, the displacement is chosen as the stochastic prior and updated to fit the measurement data more closely. For this purpose, the statFEM framework introduces a so-called model-reality mismatch, parametrized by only three hyperparameters. This makes the inference of full-field data computationally efficient in an online stage: If the stochastic prior can be computed offline, solving the underlying partial differential equation (PDE) online is unnecessary. Compared to solving a PDE, identifying only three hyperparameters and conditioning the state on the sensor data requires much fewer computational resources. Computational efficiency is an essential requirement for online applications. It is noted that conditioning a model-based prior field on sensor data represents a variant of physics-based regression.This paper presents two contributions to the existing statFEM approach: First, we use a non-intrusive polynomial chaos method to compute the prior, enabling the use of complex mechanical models in deterministic formulations. Second, we examine the influence of prior material models (linear elastic and St.Venant Kirchhoff material with uncertain Young’s modulus) on the updated solution. We present statFEM results for 1D and 2D examples, while an extension to 3D is straightforward.

Uncertainty quantification of SSG/LRR-ω turbulence model closure coefficients

The accurate simulation of complex turbulent flow has remained a challenging issue in computational fluid dynamics and plays a crucial role in engineering practice. Due to computational limitations, the Reynolds Averaged Navier-Stokes (RANS) simulation method continues to be the preferred choice for engineers to solve practical problems. Nonetheless, there are concerns regarding the applicability of RANS models to complex flows such as those with large-scale separation or jet interaction, as recommended coefficients may not be suitable. This work aims to perform a specified analysis of parameters' uncertainty in the Speziale-Sarkar-Gatsk/Launder-Reece-Rodi (SSG/LRR)-ω turbulence model. This work employs the non-intrusive polynomial chaos (NIPC) method to establish a surrogate model between parameters and computational results for uncertainty quantification. Sensitivity analysis is conducted via the Sobol index to identify key parameters critical for different flow structures. Then, Bayesian inference is applied for calibrating parameters by leveraging calculation results and two groups of experimental data, respectively. The results show that the calibrated values for the key parameters could significantly improve prediction accuracy. Subsequently, the work also delves into the inadequacies of SSG/LRR-ω models when applied to separated flows, revealing an underestimation of Reynolds stress in the separation zone as a primary source of poor predictions. Furthermore, it is considered that using a set of parameter combinations with a larger value of C3ω after the separation of the flow will obtain better calculation results in the separation zone.

Uncertainty quantification on the influence of blade thickness deviation at different rotational speeds based on flow dissipation analysis

The impact of geometric deviation due to manufacturing on compressor performance is considerable in engineering practice. To investigate the impact of blade thickness deviation on compressor performance and flow loss at various rotational speeds, a three-dimensional steady numerical simulation on Rotor 37 was conducted. The quantification of uncertainty was accomplished using a non-intrusive polynomial chaos method. The viscous dissipation coefficient was introduced to analyze the uncertain influence of blade thickness deviation on flow loss. Based on the type of loss source, the flow field was divided into six regions, including the blade tip region, blade root region, leading edge region, trailing edge region, blade surface region, and mainstream region. The results indicate that the sensitivity of total pressure ratio to thickness deviation increases significantly with an increase in the rotational speed. Under peak efficiency conditions, the effect of blade thickness deviation on flow dissipation in leading edge region decreases initially and then increases with an increase in the rotational speed. Meanwhile, the impact on flow loss in other regions increases with the increase in the rotational speed. Under near stall conditions, the blade thickness deviation has a great impact on the flow losses in the blade tip region, leading edge region, and mainstream region at 60% design rotational speed. However, the blade tip region and trailing edge region are more noticeably affected at 100% design rotational speed. Furthermore, the quantification of standard deviation of flow losses in various regions under different rotational speeds and conditions reveals that the flow loss fluctuation in the leading edge region and mainstream region varies with changes in operating conditions and rotational speeds, but the fluctuation of flow loss in other regions is independent of the rotational speed.

Sensitivity and uncertainty analysis of the nonlinear flight dynamics system of the flexible body

Free flight of a flexible body, such as large slenderness missiles, is a complex process involving rigid-flexible coupling dynamics. The strong flexible effect and the sensitive rotational characteristics caused by large slenderness ratio provide an extreme environment for the rigid-flexible coupled system. To study the performance of nonlinear flight dynamics system of the flexible body under extreme conditions, the trajectory of javelin model with a slenderness ratio of 98.15 flying in vacuum condition is calculated, and the uncertainty and the sensitivity of the coupled system are analyzed for the first time. The Monte Carlo (MC) method and the non-intrusive polynomial chaos (NIPC) method are compared by quantifying the uncertainty in the coupled system with the random moment of inertia of principal axis (Ixx). The uncertainty in the coupled system is quantified using the second-order NIPC method. It is found that the decoupled method is sensitive to the time step due to the unsynchronization of status variables between the rotational equation and structural equation. In contrast, the coupled method is insensitive to the time step. Ixx, stiffness according to the primary modes, and the initial deformation have the largest influence on the uncertainty of the coupled system. The rotational velocity in x direction (ωx) is significantly sensitive to the random inputs. The total uncertainty region of the ωx gradually expands over time.

Opposing jets for heat flux reduction and uncertainty analysis on a V-shaped blunt leading edge

With the development of three-dimensional inward-turning inlets, the V-shaped blunt leading edge model has attracted increasing attention. In the current study, an approach of using opposing jets for heat flux reduction on the V-shaped blunt leading edge is put forward. The flow structures of the V-shaped model and the model with opposing jets are further analyzed. The mechanism of heat flux generation and heat flux reduction are pointed out. By utilizing the point-collocation non-intrusive polynomial chaos (NIPC) method, the uncertainty quantification and sensitivity analysis are carried out for the parameters of the freestream and opposing jet. When calculating the uncertainty intervals, it is recommended to use the Monto Carlo method to generate massive samples for the surrogate model instead of directly using the general confidence interval. Results show that the uncertainty for heat flux is much more severe than pressure. The most contributions of the input parameters come from the freestream Mach number Ma∞ and the freestream temperature Tinf. The heat flux reduction approach proposed in this work proves to be effective for heat flux reduction and has potential application for engineering practices.

Collaborative Robust Design Optimization of Blade Geometry and Manufacturing Tolerance for a Transonic Centrifugal Impeller

Abstract Turbomachinery components are inevitably subjected to various sources of manufacturing errors. The resultant variations in blade geometry eventually lead to performance degradation. This is especially true for transonic centrifugal compressor impellers where the complex interaction between the geometric variations and shock waves may amplify such degradation. However, relevant studies on uncertainty quantification or robust design optimization of centrifugal impellers are quite rare. The mechanisms of how the realistic manufacturing errors influence the impeller aerodynamic performance are not clear. In addition, most existing studies have considered a fixed level of manufacturing tolerance applied to the impeller blade surface, which neglects the combined effect of blade geometry and manufacturing tolerance on impeller performance. In this study, a collaborative robust design optimization was performed for a transonic centrifugal impeller considering realistic manufacturing errors. The realistic manufacturing error field was first modeled based on the measurements of 92 centrifugal impellers. With a combination of computational fluid dynamics simulation and the non-intrusive polynomial chaos method, the influence of manufacturing errors on impeller performance and flow field variations was quantified. To save computational costs for the uncertainty quantification, a dual dimensionality reduction method was proposed to reduce the dimensionality of uncertainties. Finally, blade angles and the manufacturing tolerance of the impeller were collaboratively optimized to enhance the impeller performance robustness against manufacturing errors. The results show that the impeller aerodynamic performance exhibited a downward trend in the presence of manufacturing errors. The flow mechanisms responsible for this trend were mainly associated with increased intensity of the shock waves near the inducer blade tip. Such shock waves were significantly alleviated by reducing impeller inlet blade angles without the loss of pressure rise capability. Consequently, the impeller performance robustness against manufacturing errors was enhanced with the standard deviation of polytropic efficiency being reduced by 35% at a lower anticipated manufacturing cost due to increased tolerance allowance.

An Uncertainty Analysis on Finite Difference Time-Domain Computations With Artificial Neural Networks: Improving accuracy while maintaining low computational costs

Artificial neural networks (ANNs) have appeared as a potential alternative for uncertainty quantification (UQ) in the finite difference time-domain (FDTD) computation. They are applied to build a surrogate model for the computation-intensive FDTD simulation and to bypass the numerous simulations required for UQ. However, when the surrogate model utilizes an ANN, a considerable number of data are generally required for high accuracy, and generating such large quantities of data becomes computationally prohibitive. To address this drawback, a number of adaptations for ANNs are proposed, which additionally improves the accuracy of ANNs in UQ for the FDTD computation while maintaining a low computational cost. The proposed algorithm is tested for application in bioelectromagnetics, and considerable speed up, as well as the improved accuracy of UQ, is observed compared to traditional methods such as the nonintrusive polynomial chaos (NIPC) method.

Probabilistic stability of uncertain composite plates and stochastic irregularity in their buckling mode shapes: A semi-analytical non-intrusive approach

In this study, the mechanical properties of the composite plate were considered Gaussian random fields and their effects on the buckling load and corresponding mode shapes were studied by developing a semi-analytical non-intrusive approach. The random fields were decomposed by the Karhunen-Loève method. The strains were defined based on the assumptions of the first-order and higher-order shear-deformation theories. Stochastic equations of motion were extracted using Euler–Lagrange equations. The probabilistic response space was obtained by employing the non-intrusive polynomial chaos method. Finally, the effect of spatially varying stochastic properties on the critical load of the plate and the irregularity of buckling mode shapes and their sequences were studied for the first time. Our findings showed that different shear deformation plate theories could significantly influence the reliability of thicker plates under compressive loading. It is suggested that a linear relationship exists between the mechanical properties’ variation coefficient and critical loads’ variation coefficient. Also, in modeling the plate properties as random fields, a significant stochastic irregularity is obtained in buckling mode shapes, which is crucial in practical applications.

A multi-element non-intrusive Polynomial Chaos method using agglomerative clustering based on the derivatives to study irregular and discontinuous Quantities of Interest

A non-intrusive method to get a multi-element Polynomial Chaos model is developed. This method is called ME-ACD, for Multi-Element based on Agglomerative Clustering on Derivatives. It aims at approximating a Quantity of Interest which presents discontinuities or irregularities making it difficult to be accurately approximated by standard Polynomial Chaos models. The method permits to efficiently split the parameter space and to train local polynomial models of lower degrees on every element where the local pieces of the Quantity of Interest are smoother. The algorithm is based on agglomerative clustering of the observations in a well-chosen abstract space taking into account the value of the Quantity of Interest and of its derivatives with respect to the stochastic input parameters. The same observations are used for both partitioning the space and training the local models. Several partitions of the parameter space are tested, and the one leading to local models minimising a cross-validation error is selected. Once the training observations are labelled with a class number indicating the element they are located in, a neural network classifier is trained to determine which local model to use for further evaluations. The method has proven to efficiently split the parameter space for a set of applications of moderate dimension. The piecewise chaos model is compared with the ones of a standard Polynomial Chaos non-intrusive method and of a Gradient Boosted Trees method in terms of accuracy.

Quantification of parametric uncertainty in γ-Reθ model for typical flat plate and airfoil transitional flows

Owing to the lack of physical knowledge of boundary layer transition, the γ-Reθ transition model introduces closure parameters, which increase the uncertainty of transition prediction. The objective of this work is to quantify the uncertainties of closure parameters in the quantities of interests and identify the key parameters. The six closure parameters in the uncertainty intervals are used as input variables, and the uncertainties of the output results are propagated by a stochastic expansion based on the point-collocation nonintrusive polynomial chaos method. The relative contribution of each parameter to uncertainty is evaluated by the Sobol index. The computational cases include natural and bypass transitional flows on zero-pressure-gradient flat plates, and subsonic and transonic flows around airfoils. For most cases, ce2, ca2, and ca1 dominate the uncertainty, and the influence of σθt is also significant when the history effects of flow are evident. The contribution of parameters in airfoils is more complex than that in flat plates. The transonic airfoil case shows that flow separation dramatically changes the distribution of Sobol indices, which poses a challenge to the accurate prediction of transition. Generally, ce2 and ca2 are the key parameters of the γ-Reθ model.

Uncertainty optimization design of airfoil based on adaptive point adding strategy

A new and effective uncertainty optimization design framework based on adaptive point adding strategy is proposed to design and optimize the supercritical airfoil which is sensitive to Mach number variation. The new framework combines non-intrusive polynomial chaos (NIPC) method, Kriging model, lower confidence bounding (LCB) infill criterion, and global GlobalSearch algorithm. NIPC method is used to quantify the uncertainty of Mach number. A new class/shape function transformation method is used to parameterize the airfoil shapes and Kriging model is selected to establish the surrogate model of the aerodynamic optimization design. The LCB infill criterion is proposed to update the initial surrogate model by adding the optimal design point. The global GlobalSearch algorithm is used to optimize the surrogate model. The uncertainty optimization results are compared with the deterministic optimization results. It is shown that the proposed uncertainty optimization design based on adaptive point adding strategy has higher robustness under condition of Mach number variation, the optimization expense can be reduced to 20.76%, and the efficiency of optimization design is improved.

Uncertainty analysis of turbulence model in capturing flows involving laminarization and retransition

Flows experiencing laminarization and retransition are universal and crucial in many engineering applications. The objective of this study is to conduct an uncertainty quantification and sensitivity analysis of turbulence model closure coefficients in capturing laminarization and retransition for a rapidly contracting channel flow. Specifically, two commonly used turbulence models are considered: the Spalart-Allmaras (SA) one-equation model and the Menter Shear Stress Transport (SST) two-equation model. Thereby, a series of steady Reynolds Averaged Navier-Stokes (RANS) predictions of aero-engine intake acceleration scenarios are carried out with the purposely designed turbulence model closure coefficients. As a result, both SA and SST models fail to capture the retransition phenomenon though they achieve pretty good performance in laminarization. Using the non-intrusive polynomial chaos method, solution uncertainties in velocity, pressure, and surface friction are quantified and analyzed, which reveals that the SST model possesses much great uncertainty in the non-laminar regime, especially for the logarithmic law prediction. Besides, a sensitivity analysis is performed to identify the critical contributors to the solution uncertainty, and then the correlations between the closure coefficients and the deviations of the outputs of interest are obtained via the linear regression method. The results indicate that the diffusion-related constants are the dominant uncertainty contributors for both SA and SST models. Furthermore, the remarkably strong correlation between the critical closure coefficients and the outputs might be a good guide to recalibrate and even optimize the commonly used turbulence models.

Uncertainty and sensitivity analysis of heat transfer in hypersonic three-dimensional shock waves/turbulent boundary layer interaction flows

Hypersonic shock waves/turbulent boundary layer interaction flow can cause severe localized thermal loads, while the fluctuations in freestream conditions and inevitable mechanical vibrations can strongly influence the flow structures and heat transfer. This study aims to quantify the uncertainties in heat transfer and pressure of fin-plate configurations and find their main contributors. Freestream velocity, freestream static pressure, freestream static temperature, wall temperature, and fin angles are chosen as uncertainty sources. The point-collocation non-intrusive polynomial chaos method is employed to propagate the uncertainties. Subsequently, Sobol indices are utilized to denote the contribution of each input parameter. The uncertainty and sensitivity results are plotted along representative lines and on the entire bottom surface to obtain the key parameters of uncertainties. As a consequence, the uncertainty in heat transfer can exceed 30%, which is higher than that in pressure. Freestream velocity, freestream static pressure, and fin angles are the key parameters of heat transfer. Freestream velocity is the biggest contributor to the uncertainty in heat flux near the heat flux peak by influencing flow reattachment. Fin angles dominate the developments of primary vortexes and are the source of the uncertainties near the leading edge of fins. The Sobol index of freestream static pressure is relatively high at the separation line, which reveals that the boundary of flow separation is sensitive to static pressure.

Uncertainty Analysis of Parameters in SST Turbulence Model for Shock Wave-Boundary Layer Interaction

Shock wave-boundary layer interactions (SWBLIs) have a tremendous influence on the performance of hypersonic vehicles. For the numerical simulation of such engineering flows, Reynolds averaged Navier-Stokes (RANS) still occupies an irreplaceable role. However, parameters of turbulence models in RANS have substantial uncertainties, which impact the reliability of simulation results. Thus, the aim of the present study is to conduct an uncertainty analysis on parameters in the shear-stress transport (SST) turbulence model for the simulation of SWBLIs. In the current work, uncertainty quantification was performed first. A surrogate model was constructed by the non-intrusive polynomial chaos (NIPC) method to propagate uncertainties from model parameters to the quantities of interests (QoIs) and quantify them. In the subsequent sensitivity analysis, the key parameters were identified for such flow by calculating the Sobol index of each parameter for various QoIs. The results indicate that uncertainties of model parameters led to non-negligible uncertainties in those QoIs, particularly in skin friction and wall heat flux. The parameters α1, σω1, β1 were identified as primary contributors through the sensitivity analysis. Moreover, the specific effects of the three parameters on the flow prediction were analyzed by changing the parameters’ values separately.